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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 set "baseuri" "cic:/matita/RELATIONAL/NPlus/monoid".
17 include "NPlus/fun.ma".
19 (* Monoidal properties ******************************************************)
21 theorem nplus_zero_1: \forall q. zero + q == q.
22 intros. elim q; clear q; auto.
25 theorem nplus_succ_1: \forall p,q,r. NPlus p q r \to
26 (succ p) + q == (succ r).
27 intros. elim H; clear H q r; auto.
30 theorem nplus_comm: \forall p, q, x. (p + q == x) \to
31 \forall y. (q + p == y) \to x = y.
32 intros 4; elim H; clear H q x;
33 [ lapply linear nplus_inv_zero_1 to H1
34 | lapply linear nplus_inv_succ_1 to H3. decompose
38 theorem nplus_comm_rew: \forall p,q,r. (p + q == r) \to q + p == r.
39 intros. elim H; clear H q r; auto.
42 (* Corollaries of functional properties **************************************)
44 theorem nplus_inj_2: \forall p, q1, r. (p + q1 == r) \to
45 \forall q2. (p + q2 == r) \to q1 = q2.
49 (* Corollaries of nonoidal properties ***************************************)
51 theorem nplus_comm_1: \forall p1, q, r1. (p1 + q == r1) \to
52 \forall p2, r2. (p2 + q == r2) \to
53 \forall x. (p2 + r1 == x) \to
54 \forall y. (p1 + r2 == y) \to
56 intros 4. elim H; clear H q r1;
57 [ lapply linear nplus_inv_zero_2 to H1
58 | lapply linear nplus_inv_succ_2 to H3.
59 lapply linear nplus_inv_succ_2 to H4. decompose. subst.
60 lapply linear nplus_inv_succ_2 to H5. decompose
64 theorem nplus_comm_1_rew: \forall p1,q,r1. (p1 + q == r1) \to
65 \forall p2,r2. (p2 + q == r2) \to
66 \forall s. (p1 + r2 == s) \to (p2 + r1 == s).
67 intros 4. elim H; clear H q r1;
68 [ lapply linear nplus_inv_zero_2 to H1. subst
69 | lapply linear nplus_inv_succ_2 to H3. decompose. subst.
70 lapply linear nplus_inv_succ_2 to H4. decompose. subst
75 theorem nplus_shift_succ_sx: \forall p,q,r.
76 (p + (succ q) == r) \to (succ p) + q == r.
78 lapply linear nplus_inv_succ_2 to H as H0.
79 decompose. subst. auto new timeout=100.
82 theorem nplus_shift_succ_dx: \forall p,q,r.
83 ((succ p) + q == r) \to p + (succ q) == r.
85 lapply linear nplus_inv_succ_1 to H as H0.
86 decompose. subst. auto new timeout=100.
89 theorem nplus_trans_1: \forall p,q1,r1. (p + q1 == r1) \to
90 \forall q2,r2. (r1 + q2 == r2) \to
91 \exists q. (q1 + q2 == q) \land p + q == r2.
92 intros 2; elim q1; clear q1; intros;
93 [ lapply linear nplus_inv_zero_2 to H as H0.
95 | lapply linear nplus_inv_succ_2 to H1 as H0.
97 lapply linear nplus_inv_succ_1 to H2 as H0.
99 lapply linear H to H4, H3 as H0.
101 ]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
104 theorem nplus_trans_2: \forall p1,q,r1. (p1 + q == r1) \to
105 \forall p2,r2. (p2 + r1 == r2) \to
106 \exists p. (p1 + p2 == p) \land p + q == r2.
107 intros 2; elim q; clear q; intros;
108 [ lapply linear nplus_inv_zero_2 to H as H0.
110 | lapply linear nplus_inv_succ_2 to H1 as H0.
112 lapply linear nplus_inv_succ_2 to H2 as H0.
114 lapply linear H to H4, H3 as H0.
116 ]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)